Abstract

Until now there has not been an accurate method for measuring the radius of curvature, R, of a short coherence-length light source, such as a short-pulse or broadband laser. We show that the easily aligned cyclic shearing interferometer (CSI) solves this problem. The CSI produces a stable fringe pattern from which R can be determined and can be used on beams with short coherence times down to 300 fs because the two beams in the interferometer follow nearly the same path. Comparison with data from a broadband XeCl laser (30-ps coherence time) confirms that the CSI performs as theory predicts.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Bass, J. S. Whittier, “Beam divergence determination and collimation using retroreflectors,” Appl. Opt. 23, 2674–2675 (1984).
    [CrossRef] [PubMed]
  2. D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 133–136.
  3. P. Langenbeck, “Improved collimation test,” Appl. Opt. 9, 2590–2593 (1970).
    [CrossRef] [PubMed]
  4. F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
    [CrossRef]
  5. M. E. Riley, M. A. Gusinow, “Laser beam divergence utilizing a lateral shearing interferometer,” Appl. Opt. 16, 2753–2756 (1977).
    [CrossRef] [PubMed]
  6. A. Cordero-Davila, J. Pedraza-Contreras, O. Cardona-Nunez, A. Cornejo-Rodriguez, “Cyclic interferometers for optical testing,” Appl. Opt. 22, 2478–2480 (1983).
    [CrossRef] [PubMed]
  7. R. S. Sirohi, M. P. Kothiyal, “Double wedged shearing interferometer for collimation test,” Appl. Opt. 26, 4054–4056 (1987).
    [CrossRef] [PubMed]
  8. D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1080–1082 (1971).
    [CrossRef]
  9. J. C. Fouéré, D. Malacara, “Focusing errors in a collimating lens or mirror: Use of a Moiré technique,” Appl. Opt. 13, 1322–1326 (1974).
    [CrossRef] [PubMed]
  10. S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
    [CrossRef]
  11. K. Patorski, S. Yokozeki, T. Suzuki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
    [CrossRef] [PubMed]
  12. P. Hariharan, “Sagnac or Michelson-Sagnac interferometer?,” Appl. Opt. 14, 2319–2321 (1975).
    [CrossRef]
  13. P. Hariharan, D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
    [CrossRef]
  14. A. J. Montgomery, “Analysis of two-tilt compensating interferometers,” J. Opt. Soc. Am. 57, 1121–1124 (1967).
    [CrossRef]
  15. O. D. D. Soares, “Analysis and alignment of cyclic interferometers,” J. Phys. E. 11, 773–776 (1978).
    [CrossRef]
  16. Y. Li, G. Eichmann, R. R. Alfano, “Pulsed-mode laser Sagnac interferometry with applications in nonlinear optics and optical switching,” Appl. Opt. 25, 209–214 (1986).
    [CrossRef] [PubMed]
  17. R. J. Wenzel, “Tilt in the triangular shearing interferometer,” J. Opt. Soc. Am. A 4, P28 (1987).
  18. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 3–12.
  19. J. L. Carlsten, J. Rifkin, D. C. MacPherson, “Spatial mode structure of stimulated Stokes emission from a Raman generator,” J. Opt. Soc. Am. B 3, 1476–1482 (1986).
    [CrossRef]
  20. D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 464–465.

1987 (2)

R. J. Wenzel, “Tilt in the triangular shearing interferometer,” J. Opt. Soc. Am. A 4, P28 (1987).

R. S. Sirohi, M. P. Kothiyal, “Double wedged shearing interferometer for collimation test,” Appl. Opt. 26, 4054–4056 (1987).
[CrossRef] [PubMed]

1986 (2)

1984 (1)

1983 (1)

1978 (2)

F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

O. D. D. Soares, “Analysis and alignment of cyclic interferometers,” J. Phys. E. 11, 773–776 (1978).
[CrossRef]

1977 (1)

1976 (1)

1975 (2)

P. Hariharan, “Sagnac or Michelson-Sagnac interferometer?,” Appl. Opt. 14, 2319–2321 (1975).
[CrossRef]

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

1974 (1)

1971 (1)

1970 (1)

1967 (1)

1960 (1)

P. Hariharan, D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
[CrossRef]

Alfano, R. R.

Bass, M.

Cardona-Nunez, O.

Carlsten, J. L.

Cordero-Davila, A.

Cornejo-Rodriguez, A.

Dickey, F. M.

F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

Eichmann, G.

Fouéré, J. C.

Gusinow, M. A.

Harder, T. M.

F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

Hariharan, P.

P. Hariharan, “Sagnac or Michelson-Sagnac interferometer?,” Appl. Opt. 14, 2319–2321 (1975).
[CrossRef]

P. Hariharan, D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
[CrossRef]

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 3–12.

Kothiyal, M. P.

Langenbeck, P.

Li, Y.

MacPherson, D. C.

Malacara, D.

J. C. Fouéré, D. Malacara, “Focusing errors in a collimating lens or mirror: Use of a Moiré technique,” Appl. Opt. 13, 1322–1326 (1974).
[CrossRef] [PubMed]

D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 133–136.

D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 464–465.

Montgomery, A. J.

Ohnishi, K.

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

Patorski, K.

K. Patorski, S. Yokozeki, T. Suzuki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
[CrossRef] [PubMed]

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

Pedraza-Contreras, J.

Rifkin, J.

Riley, M. E.

Sen, D.

P. Hariharan, D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
[CrossRef]

Silva, D. E.

Sirohi, R. S.

Soares, O. D. D.

O. D. D. Soares, “Analysis and alignment of cyclic interferometers,” J. Phys. E. 11, 773–776 (1978).
[CrossRef]

Suzuki, T.

Wenzel, R. J.

R. J. Wenzel, “Tilt in the triangular shearing interferometer,” J. Opt. Soc. Am. A 4, P28 (1987).

Whittier, J. S.

Yokozeki, S.

K. Patorski, S. Yokozeki, T. Suzuki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
[CrossRef] [PubMed]

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

Appl. Opt. (10)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

R. J. Wenzel, “Tilt in the triangular shearing interferometer,” J. Opt. Soc. Am. A 4, P28 (1987).

J. Opt. Soc. Am. B (1)

J. Phys. E. (1)

O. D. D. Soares, “Analysis and alignment of cyclic interferometers,” J. Phys. E. 11, 773–776 (1978).
[CrossRef]

J. Sci. Instrum. (1)

P. Hariharan, D. Sen, “Cyclic shearing interferometer,” J. Sci. Instrum. 37, 374–376 (1960).
[CrossRef]

Opt. Commun. (1)

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

Opt. Eng. (1)

F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

Other (3)

D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 133–136.

D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 464–465.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 3–12.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Schematic of the CSI. M1 and M2 are mirrors, and B.S. is a 50% beam splitter. Note that the two counterrotating rays shown represent the middle of the two beams whose diameters are approximately the same as the diameters of the optics. A typical fringe pattern is shown at the CSI output.

Fig. 2
Fig. 2

Schematic of the two surfaces of a collimation tester. Surface 1 and surface 2 are separated by a distance t with normals N1 and N2, respectively. The interferometer is made of a material with an index of refraction n. The origin of the fixed coordinate system was conveniently chosen to be at the point where the incident ray, S ¯, hits the first surface at an angle α with respect to the normal. δ, is the angle in which surface 2 is tilted about the x axis.

Fig. 3
Fig. 3

A top view of the collimation tester showing the incident beam, s, being traced through the interferometer (S′, S″, S‴, S1) so that the shear, s, may be found. The beams associated with S1 and S‴ will interfere to form the fringe pattern on a screen. As in Fig. 2, α is the angle between the surface normal and the incident beam in the xy plane.

Fig. 4
Fig. 4

The side and the top views of two rays exiting a generic interferometer with a coordinate system associated with each ray. E1 and E2 are the electric fields associated with the spherical wave fronts of the two rays. θ and ξ are the vertical and the horizontal tilt components between the two rays, and vs and s are the vertical and the horizontal shear components between the two rays. Int, interferometer.

Fig. 5
Fig. 5

Schematic of a typical fringe pattern showing the various important physical parameters, s and vs are the horizontal and the vertical shears, and d is the vertical distance between the fringes. ϕ is the horizontal tilt of the fringes.

Fig. 6
Fig. 6

CSI in full detail showing all the beam direction vectors, optical surface normals, and important alignment angles. N’s, unit vectors normal to the surface.

Fig. 7
Fig. 7

Method of producing horizontal shear, s. The beam splitter (B.S.) is translated perpendicular to its optical face a distance P.

Fig. 8
Fig. 8

Method of producing horizontal shear, s, by translating one of the mirrors perpendicular to its optical face a distance P′.

Fig. 9
Fig. 9

Method of producing horizontal shear, s, by rotating both mirrors an amount β in the direction shown above.

Fig. 10
Fig. 10

Alignment of the viewing screen for the purpose of computing the path-length difference between the two exiting beams. The beam’s normal, Nscreen, is parallel to S4c and on point on the screen’s plane, P 0 SCREEN, is the point where the counterclockwise beam strikes the beam splitter.

Fig. 11
Fig. 11

Experimental setup used to test the CSI. The radius of curvature of the incident wave-front was changed by moving the collimating lens a distance γ. The camera, the frame grabber, and the computer were used to acquire, store, and analyze, the fringe patterns, respectively.

Fig. 12
Fig. 12

Experimental and theoretical results plotted against the displacement, δ, of the collimating lens from its collimation point.

Fig. 13
Fig. 13

Use of a window in the long leg of the CSI to eliminate vs.

Fig. 14
Fig. 14

Compact, one-piece design of the CSI. The tilts of the reflecting surfaces must be ground to resemble a CSI made from separate pieces. A.R., antireflection.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

S = S + Γ N .
Γ refract = { [ n 2 2 n 1 2 + ( N · S ) 2 ] } 1 / 2 N · S Γ reflect = 2 ( N · S ) ,
a = a + D * S | S | .
P 0 a ¯ · N = 0 .
S = sin α x ˆ + cos α y ˆ , S 1 = sin α x ˆ cos α y ˆ , S = sin α x ˆ + ( n 2 sin 2 α ) 1 / 2 y ˆ , S = sin α x ˆ ( n 2 sin 2 α ) 1 / 2 cos ( 2 δ ) y ˆ ( n 2 sin 2 α ) 1 / 2 sin ( 2 δ ) z ˆ , S = sin α x ˆ [ 1 n 2 + ( n 2 sin 2 α ) cos 2 ( 2 δ ) ] 1 / 2 y ˆ ( n 2 sin 2 α ) 1 / 2 sin ( 2 δ ) z ˆ ,
sin θ = ( n 2 sin 2 α ) 1 / 2 sin 2 δ,
θ ( n 2 sin 2 α ) 1 / 2 2 δ .
ξ = α arctan { sin α [ 1 n 2 + ( n 2 + sin 2 α ) cos 2 ( 2 δ ) ] 1 / 2 } .
ξ 2 δ 2 tan α ( sin 2 α n 2 ) .
Q = t sin α ( n 2 sin 2 α ) 1 / 2 cos 2 δ + 1 cos 2 δ ,
s = Q cos α = t sin ( 2 α ) ( n 2 sin 2 α ) 1 / 2 ( cos 2 δ + 1 2 cos 2 δ ) ,
s t sin 2 α ( n 2 sin 2 α ) 1 / 2 .
vs t ( n 2 sin 2 α ) 1 / 2 sin 2 δ,
E 1 = E 0 exp [ i k 2 R ( z ) ( x 2 + y 2 ) i k z ] , E 2 = E 0 exp [ i k 2 R ( z ) ( x 2 + y 2 ) i k z i β ] ,
[ x y z ] = [ cos ξ 0 sin ξ 0 1 0 sin ξ 0 cos ξ ] [ 1 0 0 0 cos θ sin θ 0 sin θ cos θ ] [ x y z ] [ s vs ( ) D ] ,
[ x y z ] = [ 1 0 ξ 0 1 0 ξ 0 1 ] [ 1 0 0 0 1 θ 0 θ 1 ] [ x y z ] [ s vs ( ) D ] .
E t = E 1 + E 2 ,
I = E t * E t .
I = sin 2 { k 2 R [ x ( s ξ L + ξ R ) + y ( vs θ L + θ R ) ] + Φ 0 2 } ,
R ( L ) R ( L + D ) = R ,
d = λ θ ( 1 L R ) + vs R .
tan ϕ = s d λ R + d ξ λ ( 1 L R ) .
R = s d λ tan ϕ .
S 0 = x ˆ , S 1 c = x ˆ , S 2 c = ( 1 2 cos 2 γ cos 2 π 8 ) x ˆ cos 2 γ sin π 4 y ˆ sin 2 γ cos π 8 z ˆ , S 3 c = sin 2 γ sin π 4 x ˆ + ( cos 2 γ sin 2 γ sin π 4 ) y ˆ sin 2 γ cos π 8 z ˆ , S 4 c = S 3 c , S 1 c c = sin 2 ψ x ˆ cos 2 ψ y ˆ sin 2 ψ cos π 4 z ˆ , S 2 c c = 2 2 x ˆ + 2 2 cos 2 ψ y ˆ 2 2 sin 2 ψ z ˆ , S 3 c c = [ 1 + ( 1 + 2 2 ) γ 2 + 2 . 61313 γψ + ψ 2 ] x ˆ + ( 2 2 γ 2 + 1 . 08239 γψ ψ 2 ) y ˆ + ( 1 . 84776 γ 2 ψ + 2 . 82843 γ 2 ψ + 1 . 08239 γψ 2 + 1 . 23184 γ 3 + 0 . 942809 ψ 3 ) z ˆ , S 4 c c = ( 2 2 γ 2 + 1 . 53073 γψ + 2 ψ 2 ) x ˆ + [ 1 ( 1 + 2 2 ) γ 2 ] y ˆ + ( 1 . 84776 γ + 1 . 23184 γ 3 0 . 585786 γ 2 ψ 0 . 448342 γψ 2 + 2 . 82843 ψ 3 ) z ˆ .
θ sin θ = 2 2 ψ 3 + ( 2 + 2 ) γ 2 ψ + γψ 2 [ ( 2 2 4 ) cos π 8 + ( 4 2 4 ) sin π 8 ] .
ξ = γ 2 2 / [ 1 ( 1 + 2 2 ) γ 2 ] { ( ) 2 2 γ 2 + 2 ψ 2 + γψ [ 2 cos π 8 + 2 sin π 8 ( 1 2 2 ) ] } ÷ { 1 ( 1 + 2 2 ) γ 2 + γψ [ 2 cos π 8 ( 1 2 2 ) + 2 sin π 8 ] } .
ψ = γ [ sin π 8 ( 2 2 1 2 ) 1 2 cos π 8 ± 1 2 ( 2 + 2 ) 1 / 2 ] .
ψ = γ [ sin π 8 + 1 2 ( 2 + 2 ) 1 / 2 ] 1 . 306 γ .
θ = 2 2 ψ 3 .
s = 2 P ,
s = 4 sin π 8 P ,
s = ( 2 + 2 ) L { sin ( π 4 + 2 β ) [ 1 + tan ( π 8 + β ) ] cos ( π 4 + 2 β ) [ 1 + cot ( 3 π 8 + β ) ] } ,
Δ L = 2 . 6678 L ψ 2 .
R theory s i f 2 δ .

Metrics